Universality in the time correlations of the long-range 1d Ising model

The equilibrium and nonequilibrium properties of ferromagnetic systems may be affected by the long-range nature of the coupling interaction. Here we study the phase separation process of a one-dimensional Ising model in the presence of a power-law decaying coupling, J(r)=1/r1+σ with σ>0, and we focus on the two-time autocorrelation function C(t,tw)=⟨si(t)si(tw)⟩. We find that it obeys the scaling form C(t,tw)=f(L(tw)/L(t)), where L(t) is the typical domain size at time t, and where f(x) can only be of two types. For σ>1, when domain walls diffuse freely, f(x) falls in the nearest-neighbour (nn) universality class. Conversely, for σ≤1, when domain walls dynamics is driven, f(x) displays a new universal behavior. In particular, the so-called Fisher-Huse exponent, which characterizes the asymptotic behavior of f(x)≃x−λ for x≫1, is λ=1 in the nn universality class (σ>1) and λ=1/2 for σ≤1